Normalized defining polynomial
\( x^{8} - 3x^{7} + 64x^{6} - 192x^{5} + 1836x^{4} - 4635x^{3} + 39520x^{2} - 110850x + 565795 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1686221298140625\) \(\medspace = 3^{6}\cdot 5^{6}\cdot 23^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.05\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{3/4}5^{3/4}23^{3/4}\approx 80.05055021202112$ | ||
Ramified primes: | \(3\), \(5\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.1686221298140625.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4552}a^{6}+\frac{229}{4552}a^{5}+\frac{1045}{4552}a^{4}-\frac{775}{4552}a^{3}-\frac{2059}{4552}a^{2}+\frac{729}{2276}a-\frac{1943}{4552}$, $\frac{1}{285557347664}a^{7}+\frac{2961779}{71389336916}a^{6}-\frac{1113940305}{35694668458}a^{5}-\frac{33644557125}{71389336916}a^{4}-\frac{3078513299}{71389336916}a^{3}+\frac{103105931573}{285557347664}a^{2}+\frac{30215055775}{285557347664}a-\frac{72836877657}{285557347664}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{6}\times C_{6}$, which has order $72$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{601}{69920996}a^{7}+\frac{2709}{69920996}a^{6}+\frac{23787}{69920996}a^{5}+\frac{130509}{69920996}a^{4}+\frac{162427}{69920996}a^{3}+\frac{365709}{34960498}a^{2}+\frac{5822035}{69920996}a-\frac{13451789}{34960498}$, $\frac{9015}{125464564}a^{7}+\frac{40635}{125464564}a^{6}+\frac{725457}{125464564}a^{5}+\frac{114375}{125464564}a^{4}+\frac{13495965}{125464564}a^{3}-\frac{7417185}{62732282}a^{2}+\frac{140785065}{125464564}a-\frac{742343551}{62732282}$, $\frac{3985231}{35694668458}a^{7}+\frac{17963379}{35694668458}a^{6}-\frac{331715375}{35694668458}a^{5}+\frac{3312640039}{35694668458}a^{4}-\frac{13606355723}{35694668458}a^{3}+\frac{19555660399}{17847334229}a^{2}-\frac{32363896855}{35694668458}a+\frac{51429790346}{17847334229}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 235.623843305 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 235.623843305 \cdot 72}{2\cdot\sqrt{1686221298140625}}\cr\approx \mathstrut & 0.321946660474 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{69}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{345}) \), \(\Q(\sqrt{5}, \sqrt{69})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(23\) | 23.8.6.1 | $x^{8} - 138 x^{4} - 217948$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.69.2t1.a.a | $1$ | $ 3 \cdot 23 $ | \(\Q(\sqrt{69}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.345.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 23 $ | \(\Q(\sqrt{345}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.119025.8t5.b.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 23^{2}$ | 8.0.1686221298140625.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |